Varieties of groupoids with comprehensive free products
A variety \(V\) has comprehensive free products if, for every integer \(m\geq 3\) there is a countable algebra \(A\in V\) such that \(A\) is isomorphic to the \(V\)-free product of \(m\) copies of \(A\), but \(A\) is not isomorphic to the \(V\)-free product of \(k\) copies of \(A\) for \(1<k<m\). The authors prove that the following varieties have comprehensive free products: the variety of all semigroups, any variety of commutative semigroups containing all semilattices, the varieties of groupoids, commutative groupoids, idempotent groupoids and commutative idempotent groupoids. They point out that this question, for the variety of idempotent semigroups, is still open.
scientific article; zbMATH DE number 3975230